On Arnold’s formula for the dimension of a polynomial ring

Abstract

If R R is a commutative integral domain with quotient field K K and x 1 , … , x n {x_1}, \ldots ,{x_n} are indeterminates, then there exist θ 1 , … , θ n {\theta _1}, \ldots ,{\theta _n} in K K such that dim ⁡ R [ x 1 , … , x n ] = n + dim ⁡ R [ θ 1 , … , θ n ] \dim R[{x_1}, \ldots ,{x_n}] = n + \dim R[{\theta _1}, \ldots ,{\theta _n}] .